The authors and Wulkenhaar exhibited a new gauge theory on the Moyal space from a one-loop effective action \[ S={1\over 4} \int d^D x(F_{\mu\nu}* F_{\mu\nu}+ \Omega^{\prime 2}\{{\mathcal A}_\mu,{\mathcal A}_\nu\}2_*+\kappa{\mathcal A}_\mu*{\mathcal A}_\mu), \] (§3 (3.15)) of the \(\phi^4\)-theory with the Grosse-Wulkenhaar term \[ S= \int d\,Dx\Biggl({1\over 2}(\partial_\mu\phi)^2+ {\Omega^2\over 2}(\widetilde x_\mu\psi)^2+ {m^2\over 2}\phi^2+ \lambda\phi^{*4}\Biggr) \]
[A. de Goursac et al., Eur. Phys. J. C, Part. Fields 56, No. 2, 293–304 (2008; Zbl 1189.81214); cf. H. Grosse and R. Wulkenhaar, Commun. Math. Phys. 256, No. 2, 305–374 (2005; Zbl 1075.82005); H. Grosse and M. Wohlgenannt, Eur. Phys. J. C, Part. Fields 52, No. 2, 435–450 (2007; Zbl 1189.81217)].
In this paper, adopting the formalism of derivation-based differential calculus and \(\varepsilon\)-connection, a mathematical interpretation of the theory (3.15) as constructed as graded curvature, is given. For this purpose, in §2. Noncommutative geometry based on \(\varepsilon\)-connection, a well written review on \(\varepsilon\)-graded algebra and \(\varepsilon\)-derivations and connections is given.
Let \(\mathbb{K}\) be a field, \(\mathbb{K}^*\) its multiplicative group, and \(\Gamma\) an abelian group. Then a commutative factor \(\varepsilon\) is a map \(\varepsilon: \Gamma\times\Gamma\to\mathbb{K}^*\) satisfying \(\varepsilon(i, j)\varepsilon(j,i)= 1\), \(\varepsilon(i,j+ k)= \varepsilon(i, j)\varepsilon(i, k)\), \(\varepsilon(i+ j,k)= \varepsilon(i,k)\varepsilon(j,k)\). An \(\varepsilon\)-Lie algebra is a \(\Gamma\)-graded vector space with the bracket product satisfying \[ [a,b]_\varepsilon=- \varepsilon(|a|, |b|)[b, a]_\varepsilon. \] An associative, unital \(\Gamma\)-graded \(\mathbb{K}\)-algebra \(A^\bullet\) is called \(\varepsilon\)-graded algebra if \(A^\bullet\) endowed a commutation factor \(\varepsilon\). Then to define \([a, b]_\varepsilon= ab- \varepsilon(|a|, |b|)ba\), we have an \(\varepsilon\)-Lie algebra \(A^\bullet_{\text{Lie},\varepsilon^\bullet}\). Typical example is the algebra (and Lie algebra) of matrix valued differential forms. Then \(\varepsilon\)-derivations, differential calculus based on \(\varepsilon\)-derivations and \(\varepsilon\)-connections are explained [M. Dubois-Violette, Math. Phys. Stud. 23, 245–306 (2001; Zbl 1038.58004)].
In §3, after dealing with \(\varepsilon\)-graded commutative algebra for super-manifolds and general properties of \(\varepsilon\)-graded matrix algebras, their differential calculus and their space of \(\varepsilon\)-connections, the Moyal algebra \({\mathcal M}_\theta\) is explained (§3.4). Then taking \(A^0_\theta={\mathcal M}_\theta\), \(A^1_\theta={\mathcal M}_\theta\), and define \(A^\bullet_\theta\) as the direct sum \(A^0_\theta\oplus A^1_\theta\) with the product \[ (a,b)\cdot(c,d)= (a* c+\alpha b* d,\,a* d+ b* c), \] \(\alpha\) is a real parameter, and define \(\varepsilon(i,j)= (-1)^{ij}\), \(i,j\in\mathbb{Z}_2\), an \(\varepsilon\)-graded algebra \({\mathcal A}^\bullet_\theta\) is defined. The authors say that this definition of \({\mathcal A}^\bullet_\theta\) mimics the Langmann-Szabo duality in the scalar case [E. Langmann and R. J. Szabo, Phys. Lett., B 533, No. 1–2, 168–177 (2002; Zbl 0994.81116)]. Then adopting the general theory in §2, \(\varepsilon\)-connections for \(A^\bullet_\theta\) are studied and recovered (3.15) by computing \(\text{Tr}(|F_{\text{ad}_a,\text{ad}_b}|^2)\). Moreover, under mild assumptions, an additional scalar field coupled to gauge fields, which can be interpreted as a Higgs field, is obtained ((3.24), (3.25)). The authors say that this is somewhat similar the interpretation of covariant coordinates as Higgs fields in the context of gauge theory models on Moyal algebras [E. Cagnache et al., J. Noncommut. Geom. 5, No. 1, 39–67 (2011; Zbl 1226.81279)].